Not quite, the answer isn't that simple. (Also if you think about what the question is asking, it cannot be < 1).
1/2 a throw?
Need to think more about this.
it never is
my rationale was that P(head)= 1/2 and P(tail)= 1/2 and if you multiply it out n times it would be to the power of n. i can now see that it is clearly wrong
If you were doing that then you would try 2^n instead of that.
Yep, except to avoid ambiguity:How does one even calculate an average?
Is it:
p_m being the probability of getting n heads in a row in m throws.
I will have a closer look at this soon, the approach was quite different to mine.Alright, here we go:
First we establish that our flips can be expressed in a string of Hs and Ts representing Heads and Tails respectively. The structure of this string will have to be:
_ _ _ _ _ ...... _ T H H H H .... H
Note, that is an 'n' number of Hs in the end of this string, we need a T before it to start the "streak" of Hs in the end, the rest is arbitrary (with some restrictions**), totaling an 'm' number of digits in our string.
**The restrictions come about when the number of remaining spots to fill 'm-n-1' spots is greater than or equal to 'n'. Since this gives rise for a chance for the streak of n Hs to be established before the 'm' moves have happened.
So now we establish the necessary domains for p_m to evaluate it
The first is clear, the second is since there are only (n+1) slots that are fixed for each string (the end ones as illustrated in the first example). The rest of the slots can be filled arbitrarily.
Now the difficult one is to consider is when
It comes down to finding
And if my calculations are correct this diverges, what have I done wrong?
No, nice just means that there is an aesthetically pleasing solution. The end answer is very simple but the problem isn't that easy.
I am not quite sure what you are doing in the calculation in your third case (if you cannot find any error in your reasoning then can you explain a little clearer please?), but that calculation shouldn't be so simple. After all, it's possible for things to happen like a string containing several occurrences of n consecutive heads. (This cannot be the main problem in your working, but its an example of why things can be tricky to do this way.)Alright, here we go:
First we establish that our flips can be expressed in a string of Hs and Ts representing Heads and Tails respectively. The structure of this string will have to be:
_ _ _ _ _ ...... _ T H H H H .... H
Note, that is an 'n' number of Hs in the end of this string, we need a T before it to start the "streak" of Hs in the end, the rest is arbitrary (with some restrictions**), totaling an 'm' number of digits in our string.
**The restrictions come about when the number of remaining spots to fill 'm-n-1' spots is greater than or equal to 'n'. Since this gives rise for a chance for the streak of n Hs to be established before the 'm' moves have happened.
So now we establish the necessary domains for p_m to evaluate it
The first is clear, the second is since there are only (n+1) slots that are fixed for each string (the end ones as illustrated in the first example). The rest of the slots can be filled arbitrarily.
Now the difficult one is to consider is when
It comes down to finding
And if my calculations are correct this diverges, what have I done wrong?
How does one even calculate an average?
Is it:
p_m being the probability of getting n heads in a row in m throws.
Using this I get it diverging lol.I'm gettingat the moment. Need to check this is right then convert it to an expected average amount of throws.
Thinking about the question would it make more sense if the series converges though?
Exactly, which is why both me and Sy were questioning what we did wrong.